A family of C1 quadrilateral finite elements
Abstract
We present a novel family of C1 quadrilateral finite elements, which define global C1 spaces over a general quadrilateral mesh with vertices of arbitrary valency. The elements extend the construction by (Brenner and Sung, J. Sci. Comput., 2005), which is based on polynomial elements of tensor-product degree p≥ 6, to all degrees p ≥ 3. Thus, we call the family of C1 finite elements Brenner-Sung quadrilaterals. The proposed C1 quadrilateral can be seen as a special case of the Argyris isogeometric element of (Kapl, Sangalli and Takacs, CAGD, 2019). The quadrilateral elements possess similar degrees of freedom as the classical Argyris triangles. Just as for the Argyris triangle, we additionally impose C2 continuity at the vertices. In this paper we focus on the lower degree cases, that may be desirable for their lower computational cost and better conditioning of the basis: We consider indeed the polynomial quadrilateral of (bi-)degree~5, and the polynomial degrees p=3 and p=4 by employing a splitting into 3×3 or 2×2 polynomial pieces, respectively. The proposed elements reproduce polynomials of total degree p. We show that the space provides optimal approximation order. Due to the interpolation properties, the error bounds are local on each element. In addition, we describe the construction of a simple, local basis and give for p∈\3,4,5\ explicit formulas for the B\'ezier or B-spline coefficients of the basis functions. Numerical experiments by solving the biharmonic equation demonstrate the potential of the proposed C1 quadrilateral finite element for the numerical analysis of fourth order problems, also indicating that (for p=5) the proposed element performs comparable or in general even better than the Argyris triangle with respect to the number of degrees of freedom.